Embodiments of the invention are directed, in general, to communication systems and, more specifically, design and decoding of hexagonal constellations.
Unlike regular QAM constellations where all constellation points within a given span are occupied, the hexagonal constellation has unused constellation points. The hexagonal constellation points are in general parameterized by two integers u and v, and the (x, y) coordinate for the i-th constellation point could be expressed as:
                              (                                    x              i                        ,                          y              i                                )                =                                            E              g                                ⁢                      (                                          (                                                      x                    0                                    ,                                      y                    0                                                  )                            +                                                u                  i                                ⁡                                  (                                      1                    ,                    0                                    )                                            +                              (                                                      1                    2                                    ,                                                            3                                        2                                                  )                                      )                                              (        1        )            where (x0, y0) is a fixed perturbation that could be used to minimize the maximum energy. The minimum distance dmin between adjacent constellation points is √{square root over (Eg)} any orthogonal rotation of coordinates does not change the constellation properties. Define the constellation average energy as:
                              E          av                ⁢        •        ⁢                              E            g                          ⁢                              ∑                          i              =              1                        N                    ⁢                                          ⁢                                                                                  x                  0                                +                                  u                  i                                +                                                      v                    i                                    2                                            ,                                                y                  0                                +                                                                            v                      i                                        ⁢                                          3                                                        2                                                                                                    (        2        )            